Momentum fluxes for creeping flow into a slot (Fig. 3.B-7). An incompressible Newtonian liquid is flowing very slowly into a thin slot of thickness 2B (in the y direction) and width W (in the z direction). The mass rate of flow in the slot is w. From the results of Problem 2B.3 it can be shown that the velocity distribution within the slot is at locations not too near the inlet. In the region outside the slot the components of the velocity for creeping flow are Equations 3B.7-1 to 4 are only approximate in the region near the slot entry for both x > 0 and x < 0. Fig. 3B.7 flow of a liquid into a slot from a semi-infinite region x < 0

(a) Find the components of the convective momentum flux p_vv inside and outside the slot. 

(b) Evaluate the xx-component of pvv at x = – a, y = 0. 

(c) Evaluate the xy-component of pvv at x = – a, y = +a. 

(d) Does the total flow of kinetic energy through the plane x = – a equal the total flow of kinetic energy through the slot? 

(e) Verify that the velocity distributions given in Eqs. 3B.7-1 to 4 satisfy the relation (∆ ∙ v) = 0. 

(f) Find the normal stress τ_1.λ at the plane y = 0 and also on the solid surface at x = 0. 

(g) Find the shear stress τ_1.λ on the solid surface at x = 0. Is this result surprising? Does sketching the velocity profile v_v vs. x at some plane y = a assist in understanding the result?

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v, = 0 v, = 0 4BWP %3D TWp (x² + y?)² x*y TWp (x² + y?)² Vy v; = 0 y = B y = -B